Mathematical Inequalities
Volume 12, Number 4 (2018), 1097–1106 doi:10.7153/jmi-2018-12-84
INEQUALITIES FOR THE WEIGHTED MEAN OF r–PREINVEX FUNCTIONS ON AN INVEX SET
DAH-YANHWANG ANDSILVESTRUSEVERDRAGOMIR
(Communicated by S. Abramovich)
Abstract. In this paper, the inequalities for the weighted mean of weaklyr-preinvex functions on an invex set are established. As applications, inequalities between the two-parameter mean of weakly r-preinvex functions and extended mean values are given.
1. Introduction
The concepts of means are very important notions in mathematics. For example, some definitions of norms are often special means and have explicit geometric mean-ings [17], and have been applied infields of heat conduction, chemistry [20], electro-statics [14] and medicine [4].
Recall the power mean Mr(x,y;λ) of order r of positive numbers x,y which is defined by
Mr(x,y;λ) =
(λxr+ (1−λ)yr)1
r, ifr=0,
xλy1−λ, ifr=0, see [7].
In [15,16], Qi gave the following weighted mean values of a positive function f
defined on the interval betweenxandywith two parameters p,q∈Rand nonnegative weightw, which is not equivalent 0, by
Mw,f(p,q;x,y)
= ⎧ ⎪ ⎨ ⎪ ⎩
y
xw(t)fp(t)dt y
xw(t)fq(t)dt
1
(p−q), if(p−q)(x−y)=0,
expxyw(t)fq(t)lnf(t)dt y
xw(t)fq(t)dt
, ifp=q,x=y.
and Mw,f(p,q;x,x) = f(x). Let x,y,s∈R, and w and f be positive and integrable functions on the closed interval[x,y]. The weighted mean of order sof the function f
on[x,y]with the weight wis defined in [8] as
M[s](f,w;x,y) = ⎧ ⎪ ⎨ ⎪ ⎩
y
xw(t)fs(t)dt y
xw(t)dt 1
s
, ifs=0,
expxyw(t)lnf(t)dt xyw(t)dt, ifs=0.
Mathematics subject classification(2010): Primary 26D15, Secondary 90C25.
Keywords and phrases: Extended means, invex set, r-convex, r-preinvex, Hermite-Hadamard in-equality.
c
, Zagreb
In addition, M[s](f,w;x,x) = f(x). By takings=p−q,p,q∈R, and replacing
w(t)byw(t)fq(t)inM[s](f,w;x,y), we have thatM[p−q](f,w fq;x,y) =M
w,f(p,q;x,y). It is obvious that the weighted mean M[s](f,w;x,y) is equivalent to the generalized weighted mean valuesMw,f(p,q;x,y). Taking w(t)≡1, the mean Mw,f(p,q;x,y) re-duces to the two-parameter meanMp,q(f;a,b)of a positive function f on[a,b]which is given in [18].
The classical Hermite-Hadamard inequality for convex functions states that if f: [a,b]→R is convex, then
fa+b
2
b1
−a
b
a f(t)dt
f(a) +f(b)
2 .
In [19], Sun and Yang extend the following right hand side of Hermite-Hadamard in-equality to the weighted mean of ordersof a positiver-convex function on an interval [a,b]. They obtain more extensive results than the main results in [5,12,13,18].
THEOREM1. Let f(t)be a positive and continuous function on the interval[x,y] with continuous derivative f(t)on [x,y], let w(t) be a positive and continuous func-tion on the range J of the funcfunc-tion f(t), and let h(t) =t . Then if f is r -convex,
M[s](f,w◦f;x,y)M[s](h,whr−1;f(x),f(y)) (1.1)
for any real number s, and if f is r -concave, the inequality is reversed.
In [9], Mohan et al. introduced the definitions of invex sets and preinvex functions. In [1, 2], Antczak investigated some interesting concept of r-invex and r-preinvex functions on an invex set and gave a new method to solve nonlinear mathematical pro-gramming problems. In [10], Noor gave some Hermite-Hadamard inequality for the preinvex and log-preinvex functions. Moreover, in [21], Wasim Ui-Haq and Javed Iqbal introduced the Hermite-Hadamard inequality forr-preinvex functions. Quite recently, in [6], Hwang and Dragomir investigated weaklyr-preinvex functions on an invex set and established some Hermite-Hadamard’s inequalities for a relation of two extended means.
Recall the following definitions ofη-path on an invex set that were introduced by Antczak in [3]. Let K⊂Rnbe a nonempty set, η:K×K→Rn andu∈K. Then the setK is said to be invex atu with respect toη, if
u+λη(v,u)∈K
for everyv∈K andλ ∈[0,1]. K is said to be an invex set with respect to η, ifK is invex at eachu∈K with respect to the same function η. Forx∈K, a closed and an openη-paths joining the pointsuandx=u+η(v,u)are defined by the notation:
Pux:={u+λη(v,u):λ∈[0,1]}
and
respectively. We note that ifη(v,u) =v−u, then the setPux=Puv={λv+ (1−λ)u:
λ∈[0,1]} is the line segment with the end pointsuandv.
Let K⊂Rn be a nonempty invex set with respect to η. The class of r-preinvex functions with respect toη is introduced via power means given by Antczak in [1]. A function f :K→R+ is said to be r-preinvex with respect to η, if there is a vector-valued functionη:K×K→Rnsuch that
f(u+λη(v,u))
(λf(v)r+ (1−λ)f(u)r)1r, ifr=0,
f(v)λf(u)1−λ, ifr=0.
for every v,u∈K and λ ∈[0,1]. We note that 0-preinvex functions are logarithmic preinvex and 1-preinvex functions are preinvex functions. It is obvious that if f is
r-preinvex, then fr is a preinvex function for positiver.
A more natural idea of weakly r-preinvex with respect to η is investigated via power means given by Hwang and Dragomir, see [6]. Let K⊂Rnbe a nonempty invex set with respect to η. A function f :K→R+ is said to be weakly r-preinvex with respect toη, if there is a vector-valued functionη:K×K→Rnsuch that
f(u+λη(v,u))Mr(f(u+η(v,u)),f(u);λ)
for every v,u∈K and λ ∈[0,1]. It is clear that if f is weakly r-preinvex, then fr is weakly preinvex for positive r, if f is weakly 0-preinvex, then log◦ f is weakly preinvex, and if f is weakly 1-preinvex, then f is weakly preinvex.
Let K⊂Rn be a nonempty invex set with respect toη:K×K→Rn. A function
f:K→R is invex with respect to the sameη. If the inequality
f(u+η(v,u)) f(v)
holds for anyu,v∈K, we say that the function f satisfies the Condition D, see [22]. We note that, if f satisfies the Condition D, f is also an r-preinvex function. In [6], applying the definition of weakly r-preinvex function, Hwang and Dragomire extend the Hermite-Hadamard inequality that involves a mean of two-parameters for weakly
r-preinvex functions on an invex set.
2. Preliminary definition and lemma
In order to obtain our results, we shall introduce the following new definition re-lated to a weighted mean for two-parameters on an invex set.
DEFINITION1. Let K⊂Rn be a nonempty invex set with respect to a
vector-valued functionη:K×K→Rn and let f,w:K→R+be integrable on theη-pathPux forx=u+η(v,u)wherev,u∈K,λ∈[0,1]. Sety(λ) =u+λη(v,u). We define the weighted mean of the function f(u+λη(v,u))on[0,1]with respect toλ by
Mp,q(f,w;u,u+η(v,u)) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
1
0w(y(λ))fp(y(λ))dλ 1
0w(y(λ))fq(y(λ))dλ
1
(p−q), ifp=q,
exp01w(y(λ))fq(y(λ))lnf(y(λ))dλ 1
0w(y(λ))fq(y(λ))dλ
, ifp=q.
In the special case, q=0,Mp,0(f,w;u,u+η(v,u)) =M[p](f,w;u,u+η(v,u))is
the weighted mean of orderp of the function f on[u,u+η(v,u)]with the weightw. LetK⊂Rnbe a nonempty invex set with respect toη:K×K→Rnandv,u∈K,
λ ∈[0,1]. We say that the function η satisfies the Condition C, see [9,11], if the following two identities
(i) η(u,u+λη(v,u)) =−λη(v,u) and
(ii) η(v,u+λη(v,u)) = (1−λ)η(v,u) hold.
In [6], Hwang and Dragomir have given the following lemma for weaklyr-preinvex functions.
LEMMA1. Let K⊂Rn be a nonempty invex set with respect to η:K×K→Rn and suppose that η satisfies Condition C. Let u∈K and let f :Pux →R for every
v∈K , λ ∈[0,1] and x=u+η(v,u)∈K . Suppose that f is continuous on Pux and
is twice-differentiable on P0
ux and r0. Then f is a weakly r -preinvex function with
respect toη if and only if
r fr−2(u){(r−1)[η(v,u)T∇f(u)]2+f(u)η(v,u)T∇2f(u)η(v,u)}0
for r>0,
{η(v,u)T∇2f(u)η(v,u)f(u)−[η(v,u)T∇f(u)]2}/f2(u)0
3. Main results
In this section, we assume that K⊂Rnbe a nonempty invex set with respect to a vector-valued functionη:K×K→Rn. Applying the definition and lemma in section 2, we have the following theorem which is our main result.
THEOREM2. Let f be a weakly r -preinvex function on an invex set K with r0. Assume that f be a positive and continuous function on Paxand twice-differentiable on
P0
ax for every a,b∈K ,λ ∈[0,1]and a<x=a+η(b,a), and letη satisfy Condition
C. Let m and M be the minimum and maximum of f on Pax, respectively. Further, let
w,h be positive and continuous on[m,M]with h(x) =x, and let g1,g2:(0,∞)→R and suppose that g2 is positive and integrable on[m,M]and the ratio g1/g2 is integrable on[m,M]. If g1/g2is increasing on[m,M], then
1
0w(f(a+λη(b,a)))g1(f(a+λη(b,a)))dλ
1
0w(f(a+λη(b,a)))g2(f(a+λη(b,a)))dλ
(3.1)
f(a+η(b,a))
f(a) w(x)hr−1(x)g1(h(x))dx
f(a+η(b,a))
f(a) w(x)hr−1(x)g2(h(x))dx
for f(a)=f(a+η(b,a)); the right-hand side of(3.1)is defined by g1(f(a))/g2(f(a)) for f(a) =f(a+η(b,a)). If g1/g2is decreasing, then the inequality(3.1)is reversed.
Proof. Letφ(λ) =fr(a+λη(b,a))forr=0 andφ(λ) =lnf(a+λη(b,a))for
r=0. We give only the proof in the case ofr>0 and g1/g2 increasing. The proof in
the other case is analogous. For convenience, letψ(λ) =f(a+λη(b,a)). Since f is weaklyr-preinvex with respect toη, Lemma1gives that
φ(λ) =r f(r−2)(a){(r−1)[η(b,a)T∇f(a)]2+f(a)η(b,a)T∇2f(a)η(b,a)}
is positive.
When f(a)=f(a+η(b,a)), it is easy to see that inequality (3.1) is equivalent to 1
0w(ψ(λ))g1(ψ(λ))dλ
1
0w(ψ(λ))g2(ψ(λ))dλ
1
0w(ψ(λ))ψr−1(λ)g1(ψ(λ))ψ(λ)dλ
1
0w(ψ(λ))ψr−1(λ)g2(ψ(λ))ψ(λ)dλ.
(3.2)
Consider
I= 1
0 w(ψ(λ))g1(ψ(λ))dλ
1
0 w(ψ(μ))ψ
r−1(μ)g
2(ψ(μ))ψ(μ)dμ (3.3)
− 1
0 w(ψ(λ))g2(ψ(λ))dλ
1
0 w(ψ(μ))ψ
r−1(μ)g
1(ψ(μ))ψ(μ)dμ
= 1
0
1
0 w(ψ(λ))w(ψ(μ))g2(ψ(λ))g2(ψ(μ))ψ
r−1(μ)ψ(μ)
×gg1(ψ(λ))
2(ψ(λ))−
g1(ψ(μ)) g2(ψ(μ))
Replacingλ andμ by each other in (3.3) and adding the resulting equations we get
I=21r 1
0
1
0 w(ψ(λ))w(ψ(μ))g2(ψ(λ))g2(ψ(μ))
(ψr(μ))−(ψr(λ)) (3.4)
×gg1(ψ(λ))
2(ψ(λ))−
g1(ψ(μ)) g2(ψ(μ))
dλdμ.
If the derivativeφ(λ) = (ψr(λ))0 for allλ∈(0,1), fromφ(λ) = (ψr(λ))0,
we always have
1
r
(ψr(μ))−(ψr(λ)))g1(ψ(λ)) g2(ψ(λ))−
g1(ψ(μ)) g2(ψ(μ))
0.
From (3.4), we get I0. This implies that the inequality (3.2) holds and then (3.1) holds. If the derivative φ(λ) = (ψr(λ))0 for all λ ∈(0,1), a similar argument
givesI0 and again the inequality (3.1) holds.
Now suppose thatφ(λ) = (ψr(λ))changes sign andφ(0)<φ(1). Thenψr(0)
ψr(1)and there exists a pointα∈(0,1)such thatφ(α) = (ψr(α))=0 and(ψr(λ))
0 for all λ ∈[0,α] and(ψr(λ))0 for all λ ∈[α,1]. Therefore, there exists a
pointβ∈(α,1)such thatψ(0) =ψ(β). Thus β
0 w(ψ(λ))ψ
r−1(λ)g
1(ψ(λ))ψ(λ)dλ
= ψ(α)
ψ(0) w(ψ(λ))x r−1g
1(x)dx+
ψ(β)
ψ(α) w(ψ(λ))x r−1g
1(x)dx=0,
and, similarly, β
0 w(ψ(λ))ψ
r−1(λ)g
2(ψ(λ))ψ(λ)dλ=0.
Consequently, the inequality (3.1) is equivalent to 1
0w(ψ(λ))g1(ψ(λ))dλ
1
0w(ψ(λ))g2(ψ(λ))dλ
1
βw(ψ(λ))ψr−1(λ)g1(ψ(λ))ψ(λ)dλ
1
βw(ψ(λ))ψr−1(λ)g2(ψ(λ))ψ(λ)dλ.
(3.5)
Consider
I2=
1
0 w(ψ(λ))g1(ψ(λ))dλ
1
β w(ψ(μ))ψ r−1(μ)g
2(ψ(μ))ψ(μ)dμ
− 1
0 w(ψ(λ))g2(ψ(λ))dλ
1
β w(ψ(μ))ψ r−1(μ)g
1(ψ(μ))ψ(μ)dμ
=1r 1
0
1
β w(ψ(λ))w(ψ(μ))g2(ψ(λ))g2(ψ(μ))ψ
r−1(μ)ψ(μ)
×gg1(ψ(λ))
2(ψ(λ))−
g1(ψ(μ)) g2(ψ(μ))
Split the double integralI2 into two parts
I21=1r
β
0
1
β w(ψ(λ))w(ψ(μ))g2(ψ(λ))g2(ψ(μ))ψ
r−1(μ)ψ(μ)
×gg1(ψ(λ))
2(ψ(λ))−
g1(ψ(μ)) g2(ψ(μ))
dλdμ,
and
I22=1r
1
β 1
β w(ψ(λ))w(ψ(μ))g2(ψ(λ))g2(ψ(μ))ψ
r−1(μ)ψ(μ)
×gg1(ψ(λ))
2(ψ(λ))−
g1(ψ(μ)) g2(ψ(μ))
dλdμ.
When(λ,μ)∈[0,β]×[β,1], we haveλμand(ψr(μ))=rψr−1(μ)ψ(μ)
0 for allμ∈(β,1). Thusψ(μ)0 for allμ∈(β,1)and
g1(ψ(λ)) g2(ψ(λ))
g1(ψ(β)) g2(ψ(β))
g1(ψ(μ)) g2(ψ(μ)).
Therefore we have thatI210. By the result proved in case ofφ(λ) = (ψr(λ))0,
we can get I220. Therefore, I2=I21+I220. It follows that (3.5) and also (3.1)
holds. Finally, if the sign of the derivativeφ(λ) = (ψr(λ)) changes andψ(0)ψ(1)
a similar proof again shows that (3.1) holds.
When f(a) = f(a+η(b,a)), ψ(0) =ψ(1), and so φ(0) =φ(1). Since φ=
(ψr(λ))0,we see thatφ= (ψr(λ)) is continuous and increasing forλ ∈(0,1).
There exists a point α∈(0,1) such that (ψr(α))=0 and (ψr(λ))0 for all λ ∈
(0,α), and (ψr(λ))0 for allλ∈(α,1).Hence
g1(ψ(λ)) g2(ψ(λ))
g1(ψ(1)) g2(ψ(1)),
for allλ∈(0,1).It follows that 1
0 w(ψ(λ))g1(ψ(λ))dλ
g1(ψ(1)) g2(ψ(1))
1
0 w(ψ(λ))g2(ψ(λ))dλ.
Therefore, the inequality (3.1) is valid. This completes the proof of Theorem2. If we takeg1(x) =xp,g2(x) =xqfor real numbers p,qin Theorem2, we get the
following weighted type of the Hermite-Hadamard inequality for weakly r-preinvex functions on an invex set.
COROLLARY1. Let f be a weakly r -preinvex function on an invex set K with r
0. Assume that f be a positive and continuous function on Pax and twice-differentiable
on P0
ax for every a,b∈K ,λ∈[0,1]and a<x=a+η(b,a), and letη satisfy
let w,h be positive and continuous on[m,M] with h(x) =x, and let p and q be real number. If p−q0, then
Mp,q(f,w◦f;a,a+η(b,a))Mp,q(h,whr−1;f(a),f(a+η(b,a))) (3.6)
for f(a)=f(a+η(b,a)); the right-hand side of(3.6)is defined by f(a)p−qfor f(a) =
f(a+η(b,a)). If p−q0, then the inequality(3.6)is reversed.
Obviously, the following corollary holds if we takeq=0 in corollary1.
COROLLARY2. Suppose that the assumptions in corollary 1 hold. If the real
number p0, then
M[p](f,w◦f;a,a+η(b,a))M[p](h,whr−1;f(a),f(a+η(b,a))) (3.7)
for f(a)=f(a+η(b,a)); the right-hand side of (3.7)is defined by f(a)pfor f(a) =
f(a+η(b,a)). If p0, then the inequality(3.7)is reversed.
REMARK1. Taking p=1 in (3.7), gives a+η(b,a)
a w(f(x))f(x)dx a+η(b,a)
a w(f(x))dx
f(a+η(b,a))
f(a) w(x)xrdx f(a+η(b,a))
f(a) w(x)xr−1dx
. (3.8)
Takingw≡1, the inequality (3.8) reduces to the inequality given by Ui-Haq and Iqbal in [21]. Further, takingr=1 or r=0, the inequality (3.8) reduces to the inequality given by Noor in [10]. So the inequality (3.1) is a greater generalization of the Hermite-Hadamard inequality for weaklyr-preinvex functions on an invex set.
REMARK2. When η(b,a) =b−a in Corollary 1, it is clear that the set K is
convex, Condition C is satisfied and the function f isr-convex. If p−q0, we have
Mp,q(f,w◦f;a,b))Mp,q(h,whr−1;f(a),f(b)) (3.9)
for f(a)= f(b); the right-hand side of (3.9) is defined by f(a)p for f(a) = f(b), while if p−q0 the inequality (3.9) is reversed. We note that the (3.9) is equivalent to the following inequality
Mw◦f,f(p,q;a,b))Mwhr−1,h(p,q;f(a),f(b)).
Takingq=0 in (3.9), the inequality (3.9) reduces to (1.1) in Theorem1. So inequality (3.1) is also more extensive than the results in [5,12,13,18]
The following corollary holds if we take w≡1 in Theorem2.
COROLLARY3. Suppose that the assumptions in theorem 2 hold and w≡1. If
g1/g2 is increasing on[m,M], then
1
0g1(f(a+λη(b,a)))dλ
1
0g2(f(a+λη(b,a)))dλ
f(a+η(b,a))
f(a) xr−1g1(x)dx f(a+η(b,a))
f(a) xr−1g2(x)dx
for f(a)=f(a+η(b,a)), the right-hand side of (3.10)is defined by g1(f(a))/g2(f(a)) for f(a) = f(a+η(b,a)), while if g1/g2 is decreasing, the inequality(3.10)is re-versed.
REMARK3. The inequality (3.10) has been given in [6]. It is clear that inequality
(3.1) is a weighted type of inequality (3.10).
Acknowledgement. The authors appreciate referees for their valuable comments and careful corrections to the original version of this paper. Thefirst author is pleased to acknowledge that this research was supported by Grant No. MOST 106-2115-M-149-001 of the Ministry of Science and Technology of the Republic of China.
R E F E R E N C E S
[1] T. ANTCZAK,r-preinvexity and r-invexity in mathematical programming, Computers and Mathematics with Applications50(3–4) (2005), 551–566,doi:10.1016/j.camwa.2005.01.024.
[2] T. ANTCZAK, A new method of solving nonlinear mathematical programming
prob-lems involving r-invex functions, J. Math. Anal. Appl. 311 (1) (2005), 313–323, doi:10.1016/j.jmaa.2005.02.049.
[3] T. ANTCZAK, Mean value in invexity analysis, Nonlinear Analysis 60 (2005), 1473–1484, doi:10.1016/j.na.2004.11.005.
[4] Y. DING,Two classes of means and their applications, Shuxue de Shijian yu Renshi (Mathematics in Practice and Theory)25(2) (1995), 16–20, (Chinese).
[5] P. M. GILL, C. E. M. PEARCE ANDJ. PECARIˇ C,´ Hadamard’s inequality for r-convex functions, J. Math. Anal. App.215(1997), 461–470.
[6] D.-Y. HWANG ANDS. S. DRAGOMIR,Extensions of the Hermite-Hadamard inequality for r-preinvex functions on an invex set, Bulletin of the Australian Mathematical Society95(3) (2017), 412–423, doi:https://doi.org/10.1017/S0004972716001374.
[7] D. S. MITRINOVIC,´ Analysis Inequalities, Springer-Verlag, Berlin, 1970.
[8] D. S. MITRINOVIC, J. E. PE´ CARIˇ C AND´ A. M. FINK,Classical and new inequalities in analysis, Kluwer Academic, Dordrecht, 1993.
[9] S. R. MOHAN ANDS. K. NEOGY,On Invex Sets and Preinvex Functions, J. Math. Anal. App.189
(1995), 901–908.
[10] M. A. NOOR,Hermite-Hadamard integral inequalities for log-preinvex functions, Journal of Mathe-matical Analysis and Approximation Theory2(2) (2007), 126–131.
[11] Z. PAVIA, S. WUB AND´ V. NOVOSELACA,Important inequalities for preinvex functions, J. Nonlinear Sci. Appl.9(2016), 3575–3579.
[12] C. E. M. PEARCE ANDJ. PECARIˇ C,´ A continuous analogue and extension of Rado’s formulae for convex and concave functions, Bull. Austral. Math. Soc.53(1996), 229–233.
[13] C. E. M. PEARCE, J. PECARIˇ C,´ ANDV.SIMIˇ C,´ Stolarsky means and Hadamard’s inequality, J. Math. Anal. Appl.220(1998), 99–109.
[14] G. P ´OLYA ANDG. SZEGO,¨ Isoperimetric Inequalities in Mathematical Physics, Princeton University Press, Princeton, 1951.
[15] F. QI,Generalized weighted mean values with two parameters, Proceedings of the Royal Society of London Series A-Mathematical, Physcial and Engineering Sciences454(1978) (1998), 2723–2732. [16] F. QI,On a two-parameter family of nonhomogeneous mean values, Tamkang Journal of Mathematics
29(2) (1998), 155–163.
[17] F. QI ANDQ.-M. LUO,Refinements and extensions of an inequality, Mathematics and Informatics Quarterly9(1) (1999), 23–25.
[19] M. SUN ANDX. YANG,Inequalities for the weighted mean of r-convex functions, Proceeding of the Americal Society133(6) (2005), 1639–1646.
[20] K. TETTAMANTI, G. S ´ARKANY´ , D. KRALIK ANDˆ R. STOMFAI,Uber die ann¨aherung logarithmis-¨ cher funktionen durch algebraische funktionen, Period. Polytech. Chem. Engrg.14(1970), 99–111. [21] W. UI-HAQ ANDJ. IQBAL,Hermite-Hadamard-type inequalities for r-Preinvex functions, Journal of
Applied Mathematics2013, 2013, Article ID 126457, 5 pages,http://dx.doi.org/10.1155/ 2013/126457.
[22] X. M. YANG, X. Q. YANG ANDK. L. TEO,Characterizations and Applications of Prequasi-Invex Functions, Journal of Optimization Theory and Applications110(3) (2001), 645–668.
(Received November 19, 2017) Dah-Yan Hwang
Department of Information and Management Taipei City University of Science and Technology No. 2, Xueyuan Rd., Beitou, 112, Taipei, Taiwan e-mail:dyhuang@tpcu.edu.tw
Silvestru Sever Dragomir Mathematics, School of Engineering & Science Victoria University P. O. Box 14428, Melbourne City, MC 8001, Australia and School of Computational & Applied Mathematics University of the Witwatersrand Private Bag 3, Johannesburg 2050, South Africa e-mail:sever.dragomir@vu.edu.au
http: // rgmia. org/ dragomir
Journal of Mathematical Inequalities
